FWIW, WebPageTest already captures the console log. If you go to the screenshot view (click on the screenshot thumbnail) any console log messages should be displayed below the screen shot. They are also exposed through the API.

The non-performance stuff I mostly push to Lighthouse (which can be tested with WebPageTest). If Lighthouse doesn't already check for that it would be worth filing a github issue.

Currently, the impression given in the first results page at the top near the color-coded "First byte time", "Keep-alive", etc section for a particular page could be that everything is mostly OK, when in reality, the page may be a disaster from a user and loading point of view.